Lesson 22

Now What Can You Build?

22.1: Notice and Wonder: Dramatic Designs (5 minutes)

Warm-up

The purpose of this warm-up is to elicit the idea that designs are built from many smaller shapes, which will be useful when students construct their own designs in a later activity. While students may notice and wonder many things about these images, perpendicular lines, parallel lines, and shapes that they know how to construct are the important discussion points. This prompt gives students opportunities to see and make use of structure (MP7). The specific structures they might notice are perpendicular lines and other shapes from constructions they have performed throughout the section.

Launch

Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Action and Expression: Internalize Executive Functions. Provide students with a table to record what they notice and wonder prior to being expected to share these ideas with others. 
Supports accessibility for: Language; Organization

Student Facing

What do you notice? What do you wonder?

A blue equilateral triangle with 3 identical white circles centered atop each midpoint. The circles do not intersect each other and have a diameter about half the side length.
Regular hexagon with center cube made from 3 rhombi. Top and upper left sides of hexagon and lower rhombus are blue.
Regular hexagon formed from a vertical center rectangle and two identical pairs of triangles on each side, rotated 180 degrees. One large right triangle and one small blue obtuse triangle.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If hexagons do not come up during the conversation, ask students to discuss where they see a hexagon.

22.2: Duplicate a Design (15 minutes)

Optional activity

Students begin by examining a design. Next, they work to recreate it and record instructions for another student to make it as they work. This provides opportunities for students to practice their construction techniques while identifying geometric figures and their properties.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Distribute copies of the blackline master. Give students access to blank paper on which to create their designs. 

If students have access to dynamic geometry software, suggest that GeoGebra Geometry from Math Tools might be helpful in this activity.

Representation: Internalize Comprehension. Begin with a physical demonstration of the construction of a regular hexagon, which will provide access for students to see the connections between new situations and prior understandings. Ask students how the construction of a regular hexagon could help them recreate the design. 
Supports accessibility for: Conceptual processing; Visual-spatial processing

Student Facing

Your teacher will give you a collection of designs that all began from the construction of a regular hexagon. Choose one to use.

  1. Record any rigid motions (rotation, reflection, or translation) you see in your design.
  2. Use straightedge and compass moves to recreate the design.
  3. Write down instructions for how to construct it.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

For students who are having trouble getting started, consider pointing them toward the display of construction techniques. Tell these students that each pattern starts with a regular hexagon or a triangle, and from there, they can use diagonals to find different points of intersection around which to center circles.

Activity Synthesis

Invite a few students to share their design and explain what constructions they used to make it.

Representing, Conversing: MLR 8 Discussion Supports.Arrange students in groups of 2. Invite students to take turns reads their instructions aloud. If time allows, the listener can attempt to create the design by following the instructions as they are read. Encourage students to press each other for detailed instructions that use mathematical language. Give students an opportunity to revise and refine their written instructions. 
Design Principle(s): Optimize output (for explanation); Cultivate conversation

22.3: Make Your Own Design (15 minutes)

Optional activity

This activity invites students to use what they have learned about constructions to create their own design. It is not required for students to begin from the construction of a regular hexagon, but it might be a good starting point for students who have trouble starting.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. For example, remind students about the construction of an equilateral triangle and the construction of a regular hexagon. Ask students how these constructions could help them create their own design.
Supports accessibility for: Social-emotional skills; Conceptual processing

Student Facing

Use dynamic geometry software to create your own design.

Write down the moves you followed so someone else can recreate your design. 

If you get stuck, consider reviewing all the constructions you have done so far. For an additional challenge, include examples of rigid motions or symmetry in your design.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

If students have access to a device that can run the GeoGebra Geometry tool from Math Tools, suggest that it might be a helpful tool in this activity. 

Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. For example, remind students about the construction of an equilateral triangle and the construction of a regular hexagon. Ask students how these constructions could help them create their own design.
Supports accessibility for: Social-emotional skills; Conceptual processing

Student Facing

Use straightedge and compass moves to create a new design.

Write down the moves you followed on that same sheet of paper so someone else can recreate your design.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Construct a tessellation with rotation, reflection, and translation symmetry.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students are unsure where to start, suggest they review all the constructions they have done so far for inspiration.

Activity Synthesis

If students will not be completing the Make Their Design activity: Invite a few students to share their design and explain what constructions they used to make it.

22.4: Make Their Design (15 minutes)

Optional activity

Students have the opportunity to practice constructing, identifying congruent parts, and justifying their thinking by following the instructions for another student's design.

Launch

Invite students to trade instructions with someone who is not their partner while keeping the design secret.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Representation: Access for Perception. Ask students to read each step of their instructions aloud to their partner. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Language

Student Facing

  1. Follow the instructions to make a design.
  2. List everything in the design that is congruent. Explain how you know.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to compare their design with the person they received the instructions from.

Invite students to share what made instructions easier or harder to follow. (It's easier when each step is written out and there are labels to help orient me. It's harder when all the steps are together in one long list without labels or stopping points.)

Lesson Synthesis

Lesson Synthesis

Collect students’ designs to decorate the classroom. Tell them that all the designs come with instructions, so they can record those instructions and recreate the designs for their own artistic projects if they would like.

Recreating these patterns or creating new patterns could be an ongoing activity for students when they need something quiet to do by themselves, after an assessment, for example.

Student Lesson Summary

Student Facing

There is a deep connection between geometry and art. Using simple construction tools, it’s possible to create beautiful patterns. Precisely recording instructions for a pattern allows other people to make the same pattern and enjoy it for themselves! These same ideas can be applied in three-dimensional space to create the objects we use and appreciate every day.